distribution), where . The Bayes discriminability . RMS sd discriminability index A common approximate (i.e. sub-optimal) discriminability index that has a closed-form is to take the average of the variances, i.e. the rms of the two standard deviations: (also denoted by ). It is times the -score of the area under the receiver operating characteristic curve (AUC) of a single-criterion observer. This index is extended to general dimensions as the Mahalanobis distance using the pooled covariance, i.e. with as the common sd matrix. Average sd discriminability index Another index is , extended to general dimensions using as the common sd matrix. Comparison of the indices It has been shown that for two univariate normal distributions, , and for multivariate normal distributions, still. Thus, and underestimate the maximum discriminability of univariate normal distributions. can underestimate by a maximum of approximately 30%. At the limit of high discriminability for univariate normal distributions, converges to . These results often hold true in higher dimensions, but not always. Simpson and Fitter promoted as the best index, particularly for two-interval tasks, but Das and Geisler have shown that is the optimal discriminability in all cases, and is often a better closed-form approximation than , even for two-interval tasks. The approximate index , which uses the geometric mean of the sd's, is less than at small discriminability, but greater at large discriminability. See also Receiver operating characteristic (ROC) Summary statistics Effect size References External links Interactive signal detection theory tutorial including calculation of d′. Detection theory Signal processing Summary statistics